{"id":8241,"date":"2023-09-29T14:21:04","date_gmt":"2023-09-29T14:21:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8241"},"modified":"2024-09-12T05:05:42","modified_gmt":"2024-09-12T05:05:42","slug":"modeling-complex-scenarios-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/modeling-complex-scenarios-background-youll-need-1\/","title":{"raw":"Modeling Complex Scenarios: Background You\u2019ll Need 1","rendered":"Modeling Complex Scenarios: Background You\u2019ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Recognize and evaluate exponential functions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>Welcome to the fascinating world of exponential and logarithmic functions! These functions are not just abstract mathematical concepts; they are vital tools that help us understand and navigate the complexities of the world around us.<\/p>\r\n<h2>Exponential Functions<\/h2>\r\n<p>An <strong>exponential function<\/strong> is expressed in the form [latex]f(x)=a\u22c5b^x[\/latex], where [latex]a[\/latex] is a constant, [latex]b[\/latex] is the base of the exponential, and [latex]x[\/latex] is the exponent. In these functions, the variable [latex]x[\/latex] is in the exponent, unlike linear functions where it is the base.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>exponential function<\/h3>\r\n\r\nFor any real number [latex]x[\/latex], an exponential function is a function with the form:\r\n\r\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\r\n\r\nwhere,\r\n\r\n<ul>\r\n\t<li>[latex]a[\/latex] is a non-zero real number called the initial value and<\/li>\r\n\t<li>[latex]b[\/latex] is any positive real number such that [latex]b\u22601[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165137601478\">Which of the following equations are\u00a0<em data-effect=\"italics\">not<\/em>\u00a0exponential functions?<\/p>\r\n<ul id=\"fs-id1165135176602\">\r\n\t<li>[latex]f(x)=4^{3(x-2)}[\/latex]<\/li>\r\n\t<li>[latex]g(x)=x^3[\/latex]<\/li>\r\n\t<li>[latex]h(x)=(\\frac{1}{3})^x[\/latex]<\/li>\r\n\t<li>[latex]j(x)=(-2)^x[\/latex]<\/li>\r\n<\/ul>\r\n<p><br \/>\r\n[reveal-answer q=\"123601\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"123601\"]<\/p>\r\n<p>By definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus, [latex]g(x)=x^3[\/latex] does not represent an exponential function because the base is an independent variable. In fact, [latex]g(x)=x^3[\/latex] is a power function.<\/p>\r\n<p>Recall that the base [latex]b[\/latex] of an exponential function is always a positive constant, and [latex]b\u22601[\/latex]. Thus, [latex]j(x)=(\u22122)^x[\/latex] does not represent an exponential function because the base, [latex]\u22122[\/latex], is less than [latex]0[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3>Evaluating Exponential Functions<\/h3>\r\n<p>Why do we limit the base [latex]b[\/latex] to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:<\/p>\r\n<ul>\r\n\t<li>Let [latex]b=\u22129[\/latex] and [latex]x=\\frac{1}{2}[\/latex]. Then [latex]f(x)=f(\\frac{1}{2})=(\u22129)^\\frac{1}{2}=\\sqrt{\u22129}[\/latex], which is not a real number.<\/li>\r\n<\/ul>\r\n<p>Why do we limit the base to positive values other than [latex]1[\/latex]? Because base [latex]1[\/latex] results in the constant function. Observe what happens if the base is [latex]1[\/latex]:<\/p>\r\n<ul>\r\n\t<li>Let [latex]b=1[\/latex]. Then [latex]f(x)=1^x=1[\/latex] for any value of [latex]x[\/latex].<\/li>\r\n<\/ul>\r\n<p>To evaluate an exponential function with the form [latex]f(x)=b^x[\/latex], we simply substitute [latex]x[\/latex] with the given value, and calculate the resulting power. For example: Let [latex]f(x)=2^x[\/latex]. What is [latex]f(3)[\/latex]?<\/p>\r\n<center>[latex]\\begin{array}{rcl} f(x) &amp; = &amp; 2^x \\\\ f(3) &amp; = &amp; 2^3 &amp; \\quad \\text{Substitute } x = 3. \\\\ &amp; = &amp; 8 &amp; \\quad \\text{Evaluate the power.} \\end{array} [\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example: Let [latex]f(x)=30(2)^x[\/latex]. What is [latex]f(3)[\/latex]?<\/p>\r\n<center>[latex]\\begin{array}{rcll} f(x) &amp; = &amp; 30(2)^x &amp; \\\\ f(3) &amp; = &amp; 30(2)^3 &amp; \\quad \\text{Substitute } x = 3. \\\\ &amp; = &amp; 30(8) &amp; \\quad \\text{Simplify the power first.} \\\\ &amp; = &amp; 240 &amp; \\quad \\text{Multiply.} \\end{array} [\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Note that if the order of operations were not followed, the result would be incorrect:<\/p>\r\n<center>[latex]f(3)=30(2)^3\u226060^3=216,000[\/latex]<\/center>\r\n<section class=\"textbox questionHelp\">\r\n<p><b>How To: Evaluating Exponential Functions<\/b><\/p>\r\n<ol>\r\n\t<li>Given an exponential function, identify [latex]a[\/latex], [latex]b[\/latex], and the value of [latex]x[\/latex] you're being asked to substitute into the function.<\/li>\r\n\t<li>Replace the variable [latex]x[\/latex] in the function with the given number.<\/li>\r\n\t<li>Compute the value of [latex]b^x[\/latex]. This means raising the base [latex]b[\/latex] to the power of [latex]x[\/latex].<\/li>\r\n\t<li>If there is a coefficient [latex]a[\/latex] in front of the base, multiply the result of [latex]b^x[\/latex] by [latex]a[\/latex]. If [latex]a[\/latex] is [latex]1[\/latex], this step does not change the value.<\/li>\r\n\t<li>Simplify the expression if necessary. This could involve performing any additional multiplication or addition\/subtraction if the function has more terms.<\/li>\r\n\t<li>Verify your result by ensuring all mathematical operations have been performed correctly and that the function has been simplified fully.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Let [latex]f(x)=5(3)^x+1[\/latex]. Evaluate [latex]f(2)[\/latex] without using a calculator.<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"586760\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"586760\"]Follow the order of operations. Be sure to pay attention to the parentheses.<\/p>\r\n<center>[latex]\\begin{array}{rcll} f(x) &amp; = &amp; 5(3)^{x+1} &amp; \\\\ f(2) &amp; = &amp; 5(3)^{2+1} &amp; \\quad \\text{Substitute } x = 2. \\\\ &amp; = &amp; 5(3)^3 &amp; \\quad \\text{Add the exponents.} \\\\ &amp; = &amp; 5(27) &amp; \\quad \\text{Simplify the power.} \\\\ &amp; = &amp; 135 &amp; \\quad \\text{Multiply.} \\end{array} [\/latex]<\/center>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13913[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize and evaluate exponential functions<\/li>\n<\/ul>\n<\/section>\n<p>Welcome to the fascinating world of exponential and logarithmic functions! These functions are not just abstract mathematical concepts; they are vital tools that help us understand and navigate the complexities of the world around us.<\/p>\n<h2>Exponential Functions<\/h2>\n<p>An <strong>exponential function<\/strong> is expressed in the form [latex]f(x)=a\u22c5b^x[\/latex], where [latex]a[\/latex] is a constant, [latex]b[\/latex] is the base of the exponential, and [latex]x[\/latex] is the exponent. In these functions, the variable [latex]x[\/latex] is in the exponent, unlike linear functions where it is the base.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>exponential function<\/h3>\n<p>For any real number [latex]x[\/latex], an exponential function is a function with the form:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\n<p>where,<\/p>\n<ul>\n<li>[latex]a[\/latex] is a non-zero real number called the initial value and<\/li>\n<li>[latex]b[\/latex] is any positive real number such that [latex]b\u22601[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165137601478\">Which of the following equations are\u00a0<em data-effect=\"italics\">not<\/em>\u00a0exponential functions?<\/p>\n<ul id=\"fs-id1165135176602\">\n<li>[latex]f(x)=4^{3(x-2)}[\/latex]<\/li>\n<li>[latex]g(x)=x^3[\/latex]<\/li>\n<li>[latex]h(x)=(\\frac{1}{3})^x[\/latex]<\/li>\n<li>[latex]j(x)=(-2)^x[\/latex]<\/li>\n<\/ul>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q123601\">Show Answer<\/button><\/p>\n<div id=\"q123601\" class=\"hidden-answer\" style=\"display: none\">\n<p>By definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus, [latex]g(x)=x^3[\/latex] does not represent an exponential function because the base is an independent variable. In fact, [latex]g(x)=x^3[\/latex] is a power function.<\/p>\n<p>Recall that the base [latex]b[\/latex] of an exponential function is always a positive constant, and [latex]b\u22601[\/latex]. Thus, [latex]j(x)=(\u22122)^x[\/latex] does not represent an exponential function because the base, [latex]\u22122[\/latex], is less than [latex]0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h3>Evaluating Exponential Functions<\/h3>\n<p>Why do we limit the base [latex]b[\/latex] to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:<\/p>\n<ul>\n<li>Let [latex]b=\u22129[\/latex] and [latex]x=\\frac{1}{2}[\/latex]. Then [latex]f(x)=f(\\frac{1}{2})=(\u22129)^\\frac{1}{2}=\\sqrt{\u22129}[\/latex], which is not a real number.<\/li>\n<\/ul>\n<p>Why do we limit the base to positive values other than [latex]1[\/latex]? Because base [latex]1[\/latex] results in the constant function. Observe what happens if the base is [latex]1[\/latex]:<\/p>\n<ul>\n<li>Let [latex]b=1[\/latex]. Then [latex]f(x)=1^x=1[\/latex] for any value of [latex]x[\/latex].<\/li>\n<\/ul>\n<p>To evaluate an exponential function with the form [latex]f(x)=b^x[\/latex], we simply substitute [latex]x[\/latex] with the given value, and calculate the resulting power. For example: Let [latex]f(x)=2^x[\/latex]. What is [latex]f(3)[\/latex]?<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl} f(x) & = & 2^x \\\\ f(3) & = & 2^3 & \\quad \\text{Substitute } x = 3. \\\\ & = & 8 & \\quad \\text{Evaluate the power.} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example: Let [latex]f(x)=30(2)^x[\/latex]. What is [latex]f(3)[\/latex]?<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcll} f(x) & = & 30(2)^x & \\\\ f(3) & = & 30(2)^3 & \\quad \\text{Substitute } x = 3. \\\\ & = & 30(8) & \\quad \\text{Simplify the power first.} \\\\ & = & 240 & \\quad \\text{Multiply.} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Note that if the order of operations were not followed, the result would be incorrect:<\/p>\n<div style=\"text-align: center;\">[latex]f(3)=30(2)^3\u226060^3=216,000[\/latex]<\/div>\n<section class=\"textbox questionHelp\">\n<p><b>How To: Evaluating Exponential Functions<\/b><\/p>\n<ol>\n<li>Given an exponential function, identify [latex]a[\/latex], [latex]b[\/latex], and the value of [latex]x[\/latex] you&#8217;re being asked to substitute into the function.<\/li>\n<li>Replace the variable [latex]x[\/latex] in the function with the given number.<\/li>\n<li>Compute the value of [latex]b^x[\/latex]. This means raising the base [latex]b[\/latex] to the power of [latex]x[\/latex].<\/li>\n<li>If there is a coefficient [latex]a[\/latex] in front of the base, multiply the result of [latex]b^x[\/latex] by [latex]a[\/latex]. If [latex]a[\/latex] is [latex]1[\/latex], this step does not change the value.<\/li>\n<li>Simplify the expression if necessary. This could involve performing any additional multiplication or addition\/subtraction if the function has more terms.<\/li>\n<li>Verify your result by ensuring all mathematical operations have been performed correctly and that the function has been simplified fully.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]f(x)=5(3)^x+1[\/latex]. Evaluate [latex]f(2)[\/latex] without using a calculator.<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q586760\">Show Answer<\/button><\/p>\n<div id=\"q586760\" class=\"hidden-answer\" style=\"display: none\">Follow the order of operations. Be sure to pay attention to the parentheses.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcll} f(x) & = & 5(3)^{x+1} & \\\\ f(2) & = & 5(3)^{2+1} & \\quad \\text{Substitute } x = 2. \\\\ & = & 5(3)^3 & \\quad \\text{Add the exponents.} \\\\ & = & 5(27) & \\quad \\text{Simplify the power.} \\\\ & = & 135 & \\quad \\text{Multiply.} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13913\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13913&theme=lumen&iframe_resize_id=ohm13913&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":8093,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8241"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":18,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8241\/revisions"}],"predecessor-version":[{"id":14793,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8241\/revisions\/14793"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/8093"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8241\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8241"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8241"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8241"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8241"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}